Complex inner products複素内積ふくそないせき and unitary matricesユニタリ行列ぎょうれつ

1Introduction

The main point of this lecture is that in complex linear algebra線型代数せんけいだいすう, ordinary transpose $A^T$A^T is not compatible with the inner product内積ないせき. Complex entries have phases, so the operation compatible with inner products is the conjugate transpose共役転置きょうやくてんち $A^{*}$A^*.

2Terms and definitions

A complex inner product space複素内積空間ふくそないせきくうかん is a complex vector space equipped with a complex inner product.

A Hermitian inner productエルミート内積ないせき has conjugate symmetry and sesquilinearity. In this series, the first variable is linear and the second is conjugate-linear:

The conjugate transpose共役転置きょうやくてんち of a complex matrix $A=\left(a_{ij}\right)$A=(a_{ij}) is

The adjoint随伴ずいはん of a linear map $T$T is the map $T^{*}$T^* satisfying

For standard complex inner products, the matrix of the adjoint is $A^{*}$A^*.

A Hermitian matrixエルミート行列ぎょうれつ satisfies

A unitary matrixユニタリ行列ぎょうれつ satisfies

A normal matrix正規行列せいきぎょうれつ satisfies

3Plan

We first fix where conjugation appears in complex inner products. Then we explain why $A^{*}$A^* moves a matrix from one side of the inner product to the other. From there, unitary matrices are seen as transformations preserving length and orthogonality, while Hermitian matrices are self-adjoint transformations compatible with the inner product.

4Intuitive explanation

For a complex number $z=i$z=i, the expression $z^2=-1$z^2=-1 cannot represent squared length. Instead,

is nonnegative. This is why complex inner products use conjugation: they must measure magnitude and phase difference without destroying positivity.

5Precise explanation

5.11. The standard complex inner product

On $C^n$\mathbb C^n, this series uses

Thus

Some textbooks choose the second variable to be linear. When that convention changes, the positions of conjugates in projection coefficients and adjoint formulas also change.

5.22. Why $A^{*}$A^* is needed

For real matrices and the standard dot product,

For complex matrices, the corresponding identity is

Thus $A^{*}$A^* is the operation that moves a linear map from the left side of the inner product to the right side.

5.33. What unitary matrices preserve

If $U^{*}U=I$U^*U=I, then for all $x,y$x,y,

Therefore a unitary matrixユニタリ行列ぎょうれつ preserves the inner product, length, and orthogonality:

It is the complex analogue of a real orthogonal matrix.

5.44. Meaning of Hermitian matrices

If $A^{*}=A$A^*=A, then

The following spectral facts are a preview used fully after eigenvalues and diagonalization have been introduced. This self-adjointness implies that eigenvalues of a Hermitian matrix are real, eigenvectors for distinct eigenvalues are orthogonal, and in finite dimensions the matrix can be diagonalized by a unitary matrix.

5.55. Hermitian versus normal

Every Hermitian matrix is normal because $A^{*}=A$A^*=A implies $A^{*}A=A{A}^{*}$A^*A=AA^*. Normal matrices are the broader class that can be unitarily diagonalized over complex finite-dimensional inner product spaces.

The eigenvalue and diagonalization properties in this table are proved later in the eigenvalue and orthogonal diagonalization lectures.

6Concrete examples

For

we get

Thus $u$u and $v$v are orthogonal. Forgetting conjugation would incorrectly give $2i$2i.

For

we have $A^{*}=A$A^*=A, so $A$A is Hermitian. Its diagonal entries are real and its off-diagonal entries are complex conjugates across the diagonal.

For

we have

So $U$U is unitary; it changes the phase of the second component without changing length.

7Criteria

• In complex inner products, always check where conjugation appears.
• In this series, the first variable is linear.
• Unitary matrices preserve inner products and lengths.
• Hermitian matrices are the complex analogue of real symmetric matrices.
• Every Hermitian matrix is normal, but not every normal matrix is Hermitian.

8Scope and limitations

The adjoint depends on the inner product. For matrices with the standard complex inner product, it is $A^{*}$A^*. With a different inner product, the same array of entries can have a different adjoint representation. Unitary diagonalization of normal matrices is a finite-dimensional complex inner-product-space result; over the real numbers alone, eigenvalues may be missing.

9Final forms

10In one sentence

A Hermitian inner productエルミート内積ないせき measures length and orthogonality while respecting complex phase, and a unitary matrixユニタリ行列ぎょうれつ preserves that structure.

In complex vector spaces, conjugation is not decoration: it is what makes $⟨x,x⟩$\langle x,x\rangle real and nonnegative. Therefore the adjoint $A^{*}$A^*, not the transpose $A^T$A^T, is the operation that interacts correctly with the complex inner product内積ないせき.

A unitary matrixユニタリ行列 satisfies $U^{*}U=I$U^*U=I, so it preserves inner products, norms, orthogonality, and distances. As a later preview, normal matrices are the complex setting for unitary diagonalization; the eigenvalue and eigenbasis statements are developed after eigenvalues and diagonalization have been introduced.

11Theorem and proof: Hermitian eigenvalues are real and eigenspaces for distinct eigenvalues are orthogonal

This theorem is a bridge to the later eigenvalue theory. Here we are not developing the full theory of eigenvalues; instead, the assumptions needed for the proof are kept inside this paragraph.

For a square matrix $A$A and a nonzero vector $v$v, if $Av=\lambda v$Av=\lambda v, then $\lambda$\lambda is an eigenvalue固有値こゆうち and $v$v is an eigenvector固有ベクトル belonging to $\lambda$\lambda. The set of all eigenvectors for the same $\lambda$\lambda, together with $0$0, is the eigenspace固有空間こゆうくうかん

The only facts used in the proof are: if $v\ne 0$v\ne0 then $⟨v,v⟩>0$\langle v,v\rangle>0; Hermitian symmetry $A^{*}=A$A^*=A gives $⟨Ax,y⟩=⟨x,Ay⟩$\langle Ax,y\rangle=\langle x,Ay\rangle; and $Av=\lambda v$Av=\lambda v is the definition of an eigenvalue-eigenvector pair.

Assume $A^{*}=A$A^*=A and $Av=\lambda v$Av=\lambda v with $v\ne 0$v\ne0. Then

Since $⟨v,v⟩>0$\langle v,v\rangle>0, we get $\lambda =\stackrel{\_}{\lambda }$\lambda=\overline\lambda, so $\lambda$\lambda is real.

Now suppose $Au=\lambda u$Au=\lambda u, $Av=\mu v$Av=\mu v, and $\lambda \ne \mu$\lambda\ne\mu. Hermitian symmetry gives

The eigenvalue $\mu$\mu is real, so $\stackrel{\_}{\mu }=\mu$\overline\mu=\mu. Hence

Because $\lambda \ne \mu$\lambda\ne\mu, $⟨u,v⟩=0$\langle u,v\rangle=0. Thus eigenvectors for distinct eigenvalues are orthogonal.

This theorem explains why Hermitian matrices play the same geometric role in complex spaces that real symmetric matrices play in real spaces.

12Equivalent characterizations of unitary matrices

For a complex square matrix $U$U, the following conditions are equivalent:

• $U^{*}U=I$U^*U=I.
• $U{U}^{*}=I$UU^*=I.
• $U^{-1}=U^{*}$U^{-1}=U^*.
• The columns of $U$U form an orthonormal basis正規直交基底せいきちょっこうきてい.
• $⟨Ux,Uy⟩=⟨x,y⟩$\langle Ux,Uy\rangle=\langle x,y\rangle for all $x,y$x,y.

Thus a unitary matrixユニタリ行列ぎょうれつ can be read either as a matrix whose inverse is its conjugate transpose or as a coordinate change that preserves the inner product.

13Theorem and proof: unitary matrices preserve inner products and norms

Let $U$U be unitary, so $U^{*}U=I$U^*U=I. For all complex vectors $x,y$x,y,

With the convention used in this lecture, the first variable is linear, so the standard complex inner product is $⟨x,y⟩=y^{*}x$\langle x,y\rangle=y^*x. Therefore

Taking $y=x$y=x gives

Since both sides are nonnegative, $\parallel Ux\parallel =\parallel x\parallel$\|Ux\|=\|x\|.

14Exercise link

15Related links